How to switch the object's parent from one bone to another? (grab, hold and drop object)
You can do it with the Child Of constraint. If you don't need to drop the object at the end, here is a quick way to do it:


Give your object 2 Child Of constraints, one with the first bone as Target, the second with the second bone. Don't forget to click on the Set Inverse button so that the object stays at its place. At frame 0 create a keyframe on the second constraint's Influence at 0. Keyframe your bones.





At frame 10, move your second bone (hand) from its current position to its position next to the object, and keyframe it. At frame 9 create a keyframe on the first Influence > 1, and on the second Influence > 0.





At frame 10 create a keyframe on the first Influence > 0, and on the second Influence > 1.





At frame 20 move the bone to its new location.